QuestionQuestion

Linear Regression

Read sections 14.2, 14.5 of the textbook and your class notes.
Mathematical Statistics and Data Analysis, 3rd ed. by John Rice. To obtain full credit, please write clearly and show your reasoning.
Problem 10A: The dataset chicks was obtained from BLSS: The Berkeley Interactive Statitstical System by Abrahams and Rizzardi. Each observation corresponds to an egg (and the resulting chick) of a bird called the Snowy Plover. The data were taken at Point Reyes Bird Observatory. Column 1 contains the egg length in millimeters, Column 2 the egg breadth in millimeters, Column 3 the egg weight in grams, and Column 4 the chick weight in grams. The object is to estimate the size of the chick based on dimensions of the egg.

a) First you are going to regress chick weight on egg length. Write out the standard regression model (yi = β0 + β1xi + ei) in terms of these two variables. You may assume that ei is normally distributed in this assignment. Plot the data and comment on whether the assumptions of the model are reasonable. Find the means and SDs of both variables, as well as the correlation between them. Use the formulas you derived in HW 9 to find the slope and the intercept of the regression line. Provide units for the slope and the intercept, and write the equation of the regression line. Draw the regression line on your plot. Do not use geom smooth() or lm() for part a.
b) Now use R to regress chick weight on egg length. Check that R produces the same slope and intercept that you got in part a (hint: see lecture 29 in-class exercise on b-courses). For each of the t and F statitics in the R output, state the null and alternative hypotheses that are being tested, and state the conclusion of the test.
c) Which of the three variables egg length, egg breadth, and egg weight is most highly correlated with chick weight? Call this one the best predictor for now. Draw the scatter diagram of chick weight versus this best predictor and put the regression line through it. Draw the residual plot. Is there any noticeable heteroscedasticity?
d) If possible, construct a 95%-confidence interval for the mean weight of Snowy Plover chicks that hatch from eggs weighing 8.5 grams. (Hint: To estimate σ in the formula for the se of the regression line use the residual se output by lm().)
e) I have a Snowy Plover egg that weighs 8.5 grams. If possible, construct a 95%- prediction interval for the weight of the chick that will hatch from this egg. Its a prediction interval, rather than a confidence interval, because its trying to predict the value of a random variable instead of estimating a fixed parameter.
f) Repeat parts d and e when the egg weight is 12 grams instead of 8.5 grams. (hint: beware of extrapola- tion)

Problem 10B: Continuing problem 10A The object is still to find a good way to predict the weight of a chick given measurements on the egg, using linear regression as the only tool. The difference between this problem and Problem 10A is that now you are going to use a combination of variables to estimate the weights of the chicks.

a) Regress the weights of the chicks on the lengths and breadths of the eggs. Assess the regression. Compare it with the best simple regression you performed Problem 10A. Is one noticeably better than the other?

b) Regress egg weight on egg length and egg breadth. Assess this regression. Use this regression to explain the similarity (or difference) between the two regressions in that were compared in a .
c) Now regress the weights of the chicks on all three predictor variables: egg length, egg breadth, and egg weight. How do you reconcile the result of the F test in this regression with the results of the t-tests? Explain why this regression is not as impressive as either of the two you compared in a, even though it has a higher R2.
d) Perform all possible regressions of chick weight using combinations of the three predictor variables used in c. Do not turn in all the results. Are there any that are clearly better than the others? Which ones, and why?

Problem 10C: This problem concerns the dataset tox.   The data are observations on a simple random sample of Hodgkins disease patients at Stanford Hospital, taken as part of a study of the toxicity of the treatment to the patients lungs.
Column 1 is the patients height in centimeters.
Column 2 is a measure of the amount of radiation to the patients lungs. Column 3 is a measure of the amount of chemotherapy the patient received.
Column 4 is a score that measures how well the patients lungs were doing before the start of treatment (also called a baseline score). Large scores are good.
Column 5 is a score that measures how well the patients lungs were doing 15 months after treatment. Large scores are good.

a) Is the treatment toxic in the short term? That is, are patients lungs worse 15 months after treatment than they were before the start of the treatment? Answer this question by carrying out statistical tests, both parametric and non-parametric. State your null and alternative hypotheses in detail, and justify your choice of procedure. [Note: the longterm effects are not very severe. The study shows that three years after treatment the patients lungs are almost back to normal.]
b) Use linear regression as your tool to decide which combination of variables in Columns 1 through 4 should be used to predict a patients score 15 months after treatment. Justify your answer.

Problem 10D: The dataset baby contains observations on mothers and their newborns at Kaiser Hospital (data courtesy of D. Nolan).
Column 1: babys weight at birth, to the nearest ounce
Column 2: gestation days (that is, total number of days of pregnancy) Column 3: mothers age in completed years
Column 4: mothers height to the nearest inch
Column 5: mothers pregnancy weight to the nearest pound
Column 6: indicator of whether the mother smoked (1) or not (0) during her pregnancy
a) Assess the normality of the babies birthweights by looking at the histogram and the normal q-q plot.
b) Draw the histogram of the mothers pregnancy weights, and draw the normal q-q plot. What feature of the q-q plot corresponds to the skewness of the distribution? What would the q-q plot look like if the skewness were in the other direction?
c) Which subset of the variables in Columns 2 through 6 should be used as predictors for birthweight? Justify your answer.
d) Interpret the coefficient of the indicator variable. How can it be used? What conclusion does it allow you to make in the present case?

Problem 10E: The dataset women contains the average weight in pounds (Column 2) for American women whose heights, correct to the nearest inch, are given in Column 1.

a) Perform the linear regression of weight on height. What is the value of R? Assess the regression.
b) Suppose the data consisted of the heights and weights of American women. That is, suppose there was a point for each woman, with one coordinate representing her weight and the other her height. Would the correlation be the same as that in a, or more, or less? Justify your choice.
c) Fit a polynomial model to the data in women. Justify your choice of degree, and assess the fit of your model.

Problem 10F: 14.52

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These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice. Unethical use is strictly forbidden.

#1) First you are going to regress chick weight on egg length. Write out the standard regression model in terms of these two variables. You may assume that ei is normally distributed in
# this assignment. Plot the data and comment on whether the assumptions of the model are reasonable.
# Find the means and SDs of both variables, as well as the correlation between them. Use the formulas
# you derived in HW 9 to find the slope and the intercept of the regression line. Provide units for the
# slope and the intercept, and write the equation of the regression line. Draw the regression line on your
# plot. Do not use geom smooth() or lm() for part a.

# Writing Function to fit a simple linear regression
simple.linear.regression = function(predictor, target)
{
sample.size = length(target)

mean.predictor = mean(predictor)
mean.target = mean(target)

sxx = sum((predictor - mean.predictor)^2)
sxy = sum((predictor - mean.predictor)*(target - mean.target))

beta1.hat = sxy/sxx
beta0.hat = mean.target - beta1.hat*mean.predictor

target.fitted = beta0.hat + beta1.hat*predictor
target.residuals = target - target.fitted
sum.squared.residuals = sum(target.residuals^2)

degree.freedom = sample.size - 2
mean.squared.error = sum.squared.residuals/degree.freedom
standard.error = sqrt(mean.squared.error)

beta0.hat.sample.variance = mean.squared.error*(1/sample.size + mean.predictor^2/sxx)
beta0.hat.standard.error = sqrt(beta0.hat.sample.variance)

beta1.hat.sample.variance = mean.squared.error/sxx
beta1.hat.standard.error = sqrt(beta1.hat.sample.variance)

t.beta0.hat = beta0.hat/beta0.hat.standard.error
t.beta1.hat = beta1.hat/beta1.hat.standard.error

p.value.beta0.hat = 2*pt(abs(t.beta0.hat), degree.freedom, lower.tail = F)
p.value.beta1.hat = 2*pt(abs(t.beta1.hat), degree.freedom, lower.tail = F)...

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